268 CHAPTER 2 Graphs and Functions If a function is not continuous at a point, then it has a discontinuity there. Figure 54 shows the graph of a function with a discontinuity at the point where x = 2. x y 0 2 The function is discontinuous at x = 2. Figure 54 EXAMPLE 1 Determining Intervals of Continuity Describe the intervals of continuity for each function in Figure 55. Figure 55 x y 0 (a) x y 0 3 (b) SOLUTION The function in Figure 55(a) is continuous over its entire domain, 1-∞, ∞2. The function in Figure 55(b) has a point of discontinuity at x = 3. Thus, it is continuous over the intervals 1-∞, 32 and 13, ∞2. S Now Try Exercises 11 and 15. Graphs of the basic functions studied in college algebra can be sketched by careful point plotting or generated by a graphing calculator. As you become more familiar with these graphs, you should be able to provide quick rough sketches of them. Identity Function ƒ1x2 =x Domain: 1-∞, ∞2 Range: 1-∞, ∞2 x y -2 -2 -1 -1 0 0 1 1 2 2 • ƒ1x2 = x increases on its entire domain, 1-∞, ∞2. • It is continuous on its entire domain, 1-∞, ∞2. x y 0 2 2 f(x) = x −10 −10 10 10 f(x) = x Figure 56 The Identity, Squaring, and Cubing Functions The identity function ƒ1x2 = x pairs every real number with itself. See Figure 56.
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